The corresponding \(D_i\)  version of the inequalities is Lemma 3.9 from [1]}. The first inequality now follows from the bound \(\Vert \tilde{D}_i\Vert _\infty \le \Vert D_i\Vert _\infty \) . The proof of the second bound in [1]} uses the bound
\(\left|\int _{Q_{n,k}} D_m(w) D_n(w) \,\mathrm {d}w\right|\lesssim A^{-4} \rho ^{m-n} \mathcal {H}^3(Q_{n,k})\) 
for \(n\ge m\) .
Let \(0\le i\le j\) . For each \(1\le k\le k_j\) , we consider two cases. If \(Q_{j,k}\in \mathcal {S}_j\) , then \(\tilde{D}_j = 0\)  on \(Q_{j,k}\) , so \(\int _{Q_{j,k}} \tilde{D}_i \tilde{D}_j \,\mathrm {d}w = 0\) . Otherwise, if \(Q_{j,k}\notin \mathcal {S}_j\) , then \(\operatorname{int}Q_{j,k} \cap S_i = \operatorname{int}Q_{j,k} \cap S_j = \emptyset \) , so \(\tilde{D}_i = D_i\)  and \(\tilde{D}_j = D_j\)  on \(Q_{j,k}\) . Therefore,
\(\left|\int _{Q_{j,k}} \tilde{D}_i(w) \tilde{D}_j(w) \,\mathrm {d}w\right|\lesssim A^{-4} \rho ^{j-i} \mathcal {H}^3(Q_{j,k}).\) 
Since the \(Q_{j,k}\) 's partition \(U\) , we sum this inequality over \(k\)  to obtain (REF ).
